Gate lock

What about a plastic anchor and screw 

Wooden matchsticks work well.

Aha;

 Believe it or not , so do bamboo skewers !     T.B.

Another dowel method is to use a dowel substantially larger than the existing hole.  Use a drill to drill a hole matching the larger dowel. Then, glue the dowel in the hole.

You do want to put the new screw in the center of the dowel, so I would suggest a pilot hole drilled as close to the center of the dowel as possible.

How about JB Weld Quick Set. Fill the hole with JB W and put in the screws and it cures in about 10 minutes. It is like steel after 24 hours.

Just realized I never updated how it went.  I used a two-part epoxy and a few toothpicks.  Went better than I thought it would.  That thing is stronger than steel now.   Thank you for all the great advice.

I even used a toothpick on the screw a cable cover plate that was warped.  My favorite new technique

Hi;

  Since I couldn't afford greens fees anymore and sold my clubs , I found that that's a good use for tees .   T.B.